Compact finite-difference (FD) schemes specify derivative approximations implicitly, thus to achieve parallelism with domain-decomposition suitable partitioning of linear systems is required. Consistent order of accuracy, dispersion, and dissipation is crucial to maintain in wave propagation problems such that deformation of the associated spectra of the discretized problems is not too severe. In this work we consider numerically tuning spectral error, at fixed formal order of accuracy to automatically devise new compact FD schemes. Grid convergence tests indicate error reduction of at least an order of magnitude over standard FD. A proposed hybrid matching-communication strategy maintains the aforementioned properties under domain-decomposition. Under evolution of linear wave-propagation problems utilizing exponential integration or explicit Runge-Kutta methods improvement is found to remain robust. A first demonstration that compact FD methods may be applied to the Z4c formulation of numerical relativity is provided where we couple our header-only, templated C++ implementation to the highly performant GR-Athena++ code. Evolving Z4c on test-bed problems shows at least an order in magnitude reduction in phase error compared to FD for propagated metric components. Stable binary-black-hole evolution utilizing compact FD together with improved convergence is also demonstrated.
翻译:常规的有限差异(FD)计划隐含地指定衍生物近似值,从而实现与域分解的平行,适合线性系统分离的线性系统需要一致的精确、分散和散射顺序,这对于维持波传播问题至关重要,这样,分解问题相关光谱的变形不会太严重。在这项工作中,我们考虑按固定的正式精确顺序对光谱错误进行数字调整,以便自动设计新的紧凑FD计划。网状趋同测试级通信战略显示至少比标准的FD减少一个数量级的错误。拟议的混合匹配通信战略将上述属性维持在域分解状态下。在利用指数集成或明确的Runge-Kutta方法改进的线性波传播问题的演变中,发现仍然很稳健。提供了第一个证明,即可将缩放法方法应用于Z4c的数值相对相对性的配方配方。我们将我们仅使用头版C++的C+执行与高度性能的GR-Athena++代码相匹配。在测试台式通信战略中将上述属性维持在域分位状态下。在域内,在使用指数级递减缩缩缩缩进阶段,与SFDFD的硬推进的硬化阶段中。</s>